in which annoyance leads to clarification ..

[while grading, last week thursday]:

argh! most of my students think that every bounded sequence of real numbers converges!?! .. is this some sort of temporary insanity, caused by quizzes?

[during lecture, last week friday]:

"before we go into the bοlzano-weιerstrass theorem, there are a few things i should clear up. first of all, what's wrong with the following proof?

"since the equation
$$1 = (-1)^{2n} = (-1)^n(-1)^n$$
"holds true for all $n \in \mathbb{N}$, it follows that
$$\lim_{n \to \infty} 1 \;=\; \left( \lim_{n \to \infty} (-1)^n \right) \left( \lim_{n \to \infty} (-1)^n \right).$$
"because the right hand side limits don't exist, it follows that constants sequences such as $x_n = 1$ are actually divergent, not convergent."

hopefully they got the point. \-:

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on an unrelated note,

1. it was another busy weekend of technical details .. and other things;
2. these days I use the phrase "the following" a lot. i blame my writing habits.